In: Formal Grammar, Barcelona, August 95.

Pomset logic as an alternative categorial grammar Alain Lecomte


Christian Retor


[email protected]

[email protected]

INRIA-Lorraine & CRIN-C.N.R.S. 615, rue du jardin botanique B.P. 101 F-54 602 Villers l s Nancy cedex FRANCE

Abstract: Lambek calculus may be viewed as a fragment of linear logic, namely intuitionistic non-commutative multiplicative linear logic. As it is too restrictive to describe numerous usual linguistic phenomena, instead of extending it we extend MLL with a non-commutative connective, thus dealing with partially ordered multisets of formulae. Relying on proof net technique, our study associates words with parts of proofs, modules, and parsing is described as proving by plugging modules. Apart from avoiding spurious ambiguities, our method succeeds in obtaining a logical description of relatively free word order, head-wrapping, clitics, and extraposition (these latest two constructions are unfortunately not included, for lack of space). 0 Introduction 1 Pomset Logic

1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Origin: coherence spaces . . . . . . . . . . . 1.1.2 Structure and properties of POMset Logic . 1.1.3 Intuitive meaning of pomset logic . . . . . . 1.2 Pomset proof nets . . . . . . . . . . . . . . . . . . 1.2.1 Dening proof nets . . . . . . . . . . . . . . 1.2.2 Cut elimination . . . . . . . . . . . . . . . . 1.4 Linguistic remarks on POMset logic . . . . . . . . 1.4.1 Relation with the Lambek calculus . . . . . 1.4.2 The fail of a too nave linguistic model . . . 1.4.3 The solution: the order on axioms . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

2 2

2 2 3 3 4 4 5 7 7 7 8

2 The linguistic model based on POMset logic and an example of relatively free word order 9 2.1 Words as modules, composition as plugging 2.2 The parsing algorithm . . . . . . . . . . . . 2.3 Immediate outcomes . . . . . . . . . . . . . 2.3.1 Getting rid of spurious ambiguities . 2.3.2 Relatively free word order . . . . . . 2.3.3 Incremental strategy . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

9 10 11 11 12 12

3 Extending the scope of categorial grammar

12

4 Conclusion 5 Appendix: Proof of lemma 10

13 14

3.1 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Head wrapping: French negation in two pictures . . . . . . . . . . . . . . . 13




and Universit
Pierre Mends-France, Grenoble

1

0 Introduction

Classical systems for grammatical deduction (NL, L, LP and so on, see for instance MO94, MK94] deal with sequents relating two sequences of formulae, each of them built by a constructor (the comma) which is associative or not, commutative or not. This gives raise to systems dealing with lists (L), multisets (LP), or binary trees (NL). Flexibility is added by the use of the so-called structural modalities Moo94, Mor94]. In this paper we aim at introducing a formalism which directly processes partially ordered multisets. This formalism, which is called POMset Logic, was introduced in Ret93], and was rather thought as a link between concurrency and linear logic Ret93, Ret95, Asp91, AD94]. Such a calculus shares many properties with ordinary (commutative) Linear Logic Gir87, Tro92, Gir95] and of course with Lambek Calculus Lam58] (this latter is nothing else than Intuitionistic Multiplicative Non Commutative Linear Logic without negation Abr91, Abr94]). Among these good logical properties, let us quote the cut-elimination theorem, strong normalisation and conuence, a proofnet syntax and a coherent semantics. But it is not a conservative extension of the Lambek calculus: it does not make possible to dene directed implications, as the order involved is rather a temporal order. Therefore we have to describe the rather spatial word order by other means. This leads to a system where words and expressions behave like parts of proofs, i.e. modules, which are plugged together either directly or by using cuts. Such a technique, rst developed in Gir86] is very related to linear logic as a logic of communication. When eliminating these cuts, a net is obtained, which exactly expresses what must be the correct word order. These properties are here applied to linguistic constructions which usually hardly t in the categorial grammar setting. As examples, we expose our description of relatively free word order, headwrapping (see the example of the French negation) and but, due to lack of space, we skipped cliticisation and extra position, which, however, simply extends the solution for head wrapping represented by French negation.

1 Pomset Logic Before getting into technicalities of proof net syntax, we give the avour of pomset logic. For more details, one should have a look at Ret93, Ret94, Ret95]

1.1 Overview

1.1.1 Origin: coherence spaces

As said above, the basic idea was to introduce non-commutative features in MLL. So we had a look at coherence semantics, the closest denotational semantics for linear logic. In this setting we observed that there just exists one non-commutative multiplicative connective that we christened before . Within this semantics the generalisation of before led us to consider partially ordered product of coherence spaces. Thus our goal was to investigate the possibility of a logical calculus involving POMsets of formulae. However without extra structure on the sequents, there are just the two usual multiplicative conjunction and disjunction, hence it is a fairly sensible solution. Ret93, Ret94]. We decided to rst look at proof net syntax, because of its close relation to coherence spaces, as explained in Ret94]. 2

1.1.2 Structure and properties of POMset Logic

The language we consider is dened from atomic formulae, with the following connectives: (unary) negation (:::)?, (binary) conjunction :::
:::, (binary) disjunction :::}:::, and (binary) before ::: < :::. We consider formulae up to de Morgan laws, which extend to this language as follows:

A? )?  A (A}B )?  A?
B ? ? ? (A < B )?  A? < B ? and not B